Weights and weight spaces
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last update: 3 March 2013
Weights and weight spaces
A finite dimensional is calibrated if it has
a basis of simultaneous eigenvectors for the
In other words, is calibrated if it has a basis such that
for all in the basis and all
Weights. Let be the abelian group generated by the elements
and let
The torus can be identified with
by identifying the element
with the homomorphism given by
The symmetric group
acts on by permuting coordinates and this action induces an action of
on given by
with notation as in (1.2).
Weight spaces. Let be a finite dimensional
For each
the space of and the
generalized space are the subspaces
respectively. From the definitions,
and is calibrated if and only if
for all
If
then
In general
but we do have
This is a decomposition of into Jordan blocks for the action of
The set of weights of is the set
An element of is called a weight vector of weight
The operators. The maps
defined below are local operators on in the sense that they act on each generalized weight space
of seperately. The operators
is only defined on the generalized weight spaces such that
Proposition 3.2.
Let
be such that
and let be a finite dimensional Define
-
The map
is well defined.
-
As operators on
whenever both sides are well defined.
 |
 |
Proof. |
|
(a) Note that
is not a well defined element of or
since it is a power series and not a Laurent polynomial. Because of this we will be careful to view
only as an operator on Let us describe this operator more precisely.
The element
acts on by
times a unipotent transformation. As an operator on
is invertible since it has determinant
where
Since this determinant is nonzero
is a well defined operator on Thus the definition of
makes sense.
The operator identities
and
if now follow easily from the definition of the
and the identities in (1.4). These identities imply that maps
into
All of the operator identities in part (b) are proved by straightforward calculations of the same flavour as the calculation of
given below. We shall not give the details for the other cases.
The only one which is really tedious is the calculation for the proof of
For a more pleasant (but less elementary) proof of this identity see Proposition 2.7 in [Ram1998].
Since both
and
are well defined. Let then
|
Let Let
be a reduced word for and define
Since the satisfy the braid relations the operator
is independent of the choice of the reduced word for The operator is a well defined
operator on if
is such that for all pairs
such that
One may use the relations in (1.5) to rewrite in the form
where
are rational functions in the variables
(The functions
are analogues of the Harish-Chandra see [Mac1971, 4.1] and
[Opd1995, Theorem 5.3].) If
is such that for all pairs
such that
then the expression
is a well defined element of the Iwahori-Hecke algebra If
with
then
The following result will be crucial to the proof of Theorem 5.5. This result is due to D. Barbasch and P. Diaconis [Dia1997]
(in the case). The proof given below is a of a
proof for the case given by S. Fomin [Fom1997].
Proposition 3.6. Let be the longest element of
Let and fix
Then
where
 |
 |
Proof. |
|
Let
Then there is a such that
and
So
and
Right multiplying by and using
the relation (1.3) gives
The element
is a multiple of the minimal central idempotent in corresponding to the representation given by
for all Up to multiplication by constants, it is the unique element
in such that
for all The lemma follows by noting that the coefficients of
in and
are both 1.
|
The action of the on weight vectors will be particularly important to the proofs of the results in
later sections. Let us record the following facts.
Let be an and let
be a weight vector in of weight
Notes and References
This is an excerpt of the preprint entitled Skew shape representations are irreducible authored by Arun Ram in 1998.
Research supported in part by National Science Foundation grant DMS-9622985, and a Postdoctoral Fellowship at Mathematical Sciences Research Institute.
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